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Question : Two circles with centres A and B touch each other externally, PQ is a direct common tangent which touches the circle at P and Q. If the radii of the circles are 9 cm and 4 cm, respectively, then the length of PQ (in cm) is equal to:

Option 1: 5

Option 2: 13

Option 3: 6.5

Option 4: 12


Team Careers360 25th Jan, 2024
Answer (1)
Team Careers360 26th Jan, 2024

Correct Answer: 12


Solution :
Let $r{_1}$​ and $r{_2}$​ be the radii of the two circles, and let $d$ be the distance between their centres.
In this case:
$d=r{_1}​+r{_2}$​
Given that the radii are 18 cm and 8 cm, we have:
$d=9+4=13\ \mathrm{cm}$
Now, the direct common tangent is the line segment that joins the points of contact of the two circles. This forms a right-angled triangle with the line segment connecting the centres. The length of the direct common tangent can be found using the Pythagorean theorem.
Let $t$ be the length of the direct common tangent.
Then:
$t^2=d^2−(r{_1}​−r{_2}​)^2$
$⇒t^2=13^2−(9−4)^2$
$⇒t^2=169−5^2$
$⇒t^2=169−25$
$⇒t^2=144$
$\therefore t=12\ \mathrm{cm}$
Hence, the correct answer is 12.

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